Algebraic & Geometric Topology

The Ptolemy field of $3$–manifold representations

Stavros Garoufalidis, Matthias Goerner, and Christian Zickert

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The Ptolemy coordinates for boundary-unipotent SL(n, )–representations of a 3–manifold group were introduced by Garoufalidis, Thurston and Zickert [arXiv:1111.2828] inspired by the A–coordinates on higher Teichmüller space due to Fock and Goncharov. We define the Ptolemy field of a (generic) PSL(2, )-representation and prove that it coincides with the trace field of the representation. This gives an efficient algorithm to compute the trace field of a cusped hyperbolic manifold.

Article information

Algebr. Geom. Topol., Volume 15, Number 1 (2015), 371-397.

Received: 21 January 2014
Revised: 9 May 2014
Accepted: 7 July 2014
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57N10: Topology of general 3-manifolds [See also 57Mxx]
Secondary: 57M27: Invariants of knots and 3-manifolds

Ptolemy coordinates trace field SnapPy $3$–manifold


Garoufalidis, Stavros; Goerner, Matthias; Zickert, Christian. The Ptolemy field of $3$–manifold representations. Algebr. Geom. Topol. 15 (2015), no. 1, 371--397. doi:10.2140/agt.2015.15.371.

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  • W Bosma, J Cannon, C Playoust, The Magma algebra system, I: The user language, J. Symbolic Comput. 24 (1997) 235–265
  • D Calegari, Real places and torus bundles, Geom. Dedicata 118 (2006) 209–227
  • M Culler, N,M Dunfield, J,R Weeks, SnapPy: A computer program for studying the geometry and topology of $3$–manifolds \setbox0\makeatletter\@url {\unhbox0
  • N,M Dunfield, Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds, Invent. Math. 136 (1999) 623–657
  • E Falbel, S Garoufalidis, A Guilloux, M Görner, P-V Koseleff, F Rouillier, C Zickert, CURVE database \setbox0\makeatletter\@url {\unhbox0
  • E Falbel, P,V Koseleff, F Rouillier, Representations of fundamental groups of $3$–manifolds into $\mathrm{PGL}(3,\mathbb C)$: Exact computations in low complexity
  • V Fock, A Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. (2006) 1–211
  • S Garoufalidis, M Goerner, C,K Zickert, Gluing equations for $\rm{PGL}(n,\mathbb C)$–representations of $3$–manifolds To appear in Alg. & Geom. Topol.
  • S Garoufalidis, C,D Hodgson, H Rubinstein, H Segerman, $1$–efficient triangulations and the index of a cusped hyperbolic $3$–manifold To appear in Geom. & Topol.
  • S Garoufalidis, D,P Thurston, C,K Zickert, The complex volume of $\rm{SL}(n,\mathbb C)$–representations of $3$–manifolds To appear in Duke Math. J.
  • S Garoufalidis, C,K Zickert, The symplectic properties of the $\mathrm{PGL}(n,\mathbb C)$–gluing equations To appear in J. Quantum Topol.
  • C Maclachlan, A,W Reid, The arithmetic of hyperbolic $3$–manifolds, Graduate Texts in Mathematics 219, Springer, New York (2003)
  • W,D Neumann, Combinatorics of triangulations and the Chern–Simons invariant for hyperbolic $3$–manifolds, from: “Topology '90”, Ohio State Univ. Math. Res. Inst. Publ. 1, de Gruyter, Berlin (1992) 243–271
  • C,K Zickert, The volume and Chern–Simons invariant of a representation, Duke Math. J. 150 (2009) 489–532